Tunneling through a barrier with the phase-amplitude method

A previous study (Rawitscher, 2015) of the solution of Milne's non linear equation for the phase and amplitude of a one-dimensional wave function is extended to the case where the incident energy is less than the potential (Barrier region). The numerical method again consists in implementing a spectral expansion of the amplitude in terms of a number of Chebyshev polynomials. The method is applied to a Morse-type potential, for energies in a resonance region, and for one energy below the resonance region, and a strong repulsive Coulomb potential. The results are compared with highly accurate direct solutions of the Schrödinger equation, and were found to be accurate to 1:10−6.